Skip to main content

Stability and Bifurcation of a Soap Film Spanning a Flexible Loop


The Euler–Plateau problem, proposed by Giomi and Mahadevan in Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012), concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler–Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For a spanning surface with energy density σ and a bounding loop with length 2πR and flexural rigidity a, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at R 3 σ/a=3, confirming a result obtained previously via an energy comparison. All other bifurcation solution branches emanating from the flat circular solution branch, including those to nonplanar solution branches, are found to be unstable.

This is a preview of subscription content, access via your institution.


  1. See, for instance, Proposition 2.1 of Gurtin and Murdoch [14].


  1. Dierkes, U., Hildebrandt, S., Tromba, A.J.: Regularity of Minimal Surfaces, 2nd edn. Springer, Berlin (2010)

    MATH  Google Scholar 

  2. Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, 2nd edn. Springer, Berlin (2010)

    Book  Google Scholar 

  3. Bernatzki, F., Ye, R.: Minimal surfaces with an elastic boundary. Ann. Glob. Anal. Geom. 19, 1–9 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Giomi, L., Mahadevan, L.: Minimal surfaces bounded by elastic lines. Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012)

    ADS  Article  MathSciNet  Google Scholar 

  5. Bernatzki, F.: Mass-minimizing currents with an elastic boundary. Manuscr. Math. 93, 1–20 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bernatzki, F.: On the existence and regularity of mass-minimizing currents with an elastic boundary. Ann. Glob. Anal. Geom. 15, 379–399 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Plateau, J.A.F.: Recherches expérimentales et théorique sur les figures d’équilibre d’une masse liquide sans pesanteur. Mém. Acad. R. Sci. Lett. Beaux-Arts Belg. 23, 1–151 (1849)

    Google Scholar 

  8. Singer, D.A.: Lectures on elastic curves and rods. In: Garay, O.J., García-Río, E., Vázquez-Lorenzo, R. (eds.) Curvature and Variational Modeling in Physics and Biophysics. Conference Proceedings of the American Institute of Physics, vol. 1002, pp. 3–32 (2008)

    Google Scholar 

  9. do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976)

    MATH  Google Scholar 

  10. Efimov, N.V.: Some problems in the theory of space curves. Usp. Mat. Nauk 2, 193–194 (1947)

    MathSciNet  Google Scholar 

  11. Fenchel, W.: On the differential geometry of closed space curves. Bull. Am. Math. Soc. 57, 44–54 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  13. Chen, Y.-C.: Singularity theory and nonlinear bifurcation analysis. In: Fu, Y.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  14. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    MATH  MathSciNet  Google Scholar 

  15. Julicher, F.: Supercoiling transitions of closed DNA. Phys. Rev. E 49, 2429–2435 (1994)

    ADS  Article  Google Scholar 

  16. Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds.) Mathematical Approaches to Biomolecular Structure and Dynamics, pp. 71–113. Springer, Berlin (1996)

    Chapter  Google Scholar 

  17. Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elast. 60, 173–221 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ericksen, J.L.: The thermo-kinetic view of elastic stability theory. Int. J. Solids Struct. 2, 573–580 (1966)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Eliot Fried.

Additional information

Dedicated to Roger Fosdick.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, Yc., Fried, E. Stability and Bifurcation of a Soap Film Spanning a Flexible Loop. J Elast 116, 75–100 (2014).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Surface tension
  • Flexural rigidity
  • Inextensibility
  • Euler–Lagrange equations
  • Second variation condition
  • Plateau’s problem
  • Thread problem
  • Closed-curve problem

Mathematics Subject Classification (2010)

  • 49Q10
  • 53A04
  • 53A05
  • 53A10
  • 53A25
  • 53C80
  • 53Z05